Matrix Operations

🔢 Matrix Operations

Part 1 of 5 — Meet the Matrix

Topics in This Part

🔑 Key Concept: A matrix is just a rectangular grid of numbers. Before we add, subtract, or multiply them, we need a shared language for describing them — their size and the address of every number inside.

What Is a Matrix?

A matrix (plural matrices) is a rectangular array of numbers arranged in rows and columns, written inside brackets:

$A = \begin{bmatrix} 2 & -1 & 5 \\ 0 & 3 & 4 \end{bmatrix}$

• Rows run horizontally (left to right). $A$ has 2 rows.
• Columns run vertically (top to bottom). $A$ has 3 columns.

Each number inside is called an entry (or element).

💡 Memory trick: "Rows are Right-across, Columns are Climbing-down." Always read rows first, columns second.

Dimensions and Entries

The dimensions (or order) of a matrix are written $\text{rows} \times \text{columns}$ — always rows first.

$A = \begin{bmatrix} 2 & -1 & 5 \\ 0 & 3 & 4 \end{bmatrix} \quad\text{is a } 2 \times 3 \text{ matrix.}$

To name a single entry we use $a_{ij}$, where $i$ is the row number and $j$ is the column number:

🔑 Key Idea: The subscript order matches the dimension order — $a_{ij}$ means row $i$, column $j$.

Concept Check 🎯

Locate the Entries 🧮

Use $M = \begin{bmatrix} 9 & 2 & -4 \\ 0 & 7 & 1 \end{bmatrix}$.

1) $M$ has how many rows? 2) $M$ has how many columns? 3) What is the value of $m_{13}$?

When Are Two Matrices Equal?

Two matrices are equal only if both conditions hold:

1. They have the same dimensions, and
2. Every matching entry is identical: $a_{ij} = b_{ij}$ for all $i, j$.

Example

$\begin{bmatrix} x & 5 \\ 7 & y \end{bmatrix} = \begin{bmatrix} 3 & 5 \\ 7 & -2 \end{bmatrix} \;\Longrightarrow\; x = 3,\; y = -2$

We match the matrices position-by-position: the top-left entries must agree ($x = 3$) and the bottom-right entries must agree ($y = -2$).

⚠️ A $2 \times 3$ matrix can never equal a $3 \times 2$ matrix, even if they contain the very same numbers. Shape comes first.

You now have the vocabulary every operation depends on. In Part 2 we start combining matrices with addition, subtraction, and scalar multiplication.

Solve for the Unknowns 🔽

Given $\begin{bmatrix} x & 8 \\ -3 & y \end{bmatrix} = \begin{bmatrix} 6 & 8 \\ -3 & 1 \end{bmatrix}$, match each entry.

🔢 Matrix Operations

Part 2 of 5 — Addition, Subtraction & Scalar Multiplication

🔑 The Idea: Addition, subtraction, and multiplying by a single number all work entry-by-entry. These are the "easy" operations — matrix multiplication (Part 3) is the one with a twist.

Adding and Subtracting Matrices

To add or subtract two matrices, combine corresponding entries.

⚠️ You can only add or subtract matrices with the same dimensions. If the shapes differ, the operation is undefined.

Worked Example: Addition

$\begin{bmatrix} 2 & 1 \\ 4 & 0 \end{bmatrix} + \begin{bmatrix} 3 & -5 \\ 1 & 6 \end{bmatrix} = \begin{bmatrix} 2+3 & 1+(-5) \\ 4+1 & 0+6 \end{bmatrix} = \begin{bmatrix} 5 & -4 \\ 5 & 6 \end{bmatrix}$

Worked Example: Subtraction

$\begin{bmatrix} 7 & 2 \\ -1 & 8 \end{bmatrix} - \begin{bmatrix} 3 & 5 \\ 4 & 2 \end{bmatrix} = \begin{bmatrix} 7-3 & 2-5 \\ -1-4 & 8-2 \end{bmatrix} = \begin{bmatrix} 4 & -3 \\ -5 & 6 \end{bmatrix}$

💡 Line the two matrices up like grids and march through them position by position. Watch the signs carefully on subtraction.

Concept Check 🎯

Scalar Multiplication

A scalar is just a single number. To multiply a matrix by a scalar, multiply every entry by that scalar:

$3 \begin{bmatrix} 2 & -1 \\ 0 & 4 \end{bmatrix} = \begin{bmatrix} 3\cdot 2 & 3\cdot(-1) \\ 3\cdot 0 & 3\cdot 4 \end{bmatrix} = \begin{bmatrix} 6 & -3 \\ 0 & 12 \end{bmatrix}$

This even handles a negative scalar — multiplying by $-1$ flips every sign:

$-1 \begin{bmatrix} 5 & -2 \\ 3 & 7 \end{bmatrix} = \begin{bmatrix} -5 & 2 \\ -3 & -7 \end{bmatrix}$

🔑 Key Idea: Subtraction is really "add the opposite." $A - B = A + (-1)B$, which is why both operations need matching dimensions.

Scalar & Combined Practice 🧮

Let $A = \begin{bmatrix} 4 & -2 \\ 1 & 5 \end{bmatrix}$ and $B = \begin{bmatrix} 3 & 0 \\ -1 & 2 \end{bmatrix}$.

1) In $2A$, what is the top-right entry? 2) In $A - B$, what is the bottom-left entry? 3) In $A + 3B$, what is the top-left entry?

Properties of Addition 🔽

These operations behave just like ordinary numbers. Fill in each property (assume $A, B$ have matching dimensions).

🔢 Matrix Operations

Part 3 of 5 — Matrix Multiplication

🔑 Why it's different: Matrix multiplication is not entry-by-entry. Instead, each output entry is a row-times-column dot product. Get the setup right and the arithmetic is easy.

When Can You Multiply? The Inner-Match Rule

To multiply $A \times B$, the number of columns in $A$ must equal the number of rows in $B$.

$\underbrace{A}_{m \times \textcolor{red}{n}} \;\cdot\; \underbrace{B}_{\textcolor{red}{n} \times p} \;=\; \underbrace{C}_{m \times p}$

• The inner numbers must match → tells you if multiplication is allowed.
• The outer numbers give the size of the product.

Examples

⚠️ Because the inner numbers must match, $AB$ being defined does not mean $BA$ is defined. Order matters!

Can You Multiply? 🔽

For each pair, decide whether the product $AB$ exists and, if so, its dimensions.

How to Multiply: Row · Column

Entry $c_{ij}$ of the product comes from row $i$ of $A$ dotted with column $j$ of $B$: multiply matching pieces, then add.

Worked Example: $2 \times 2$ times $2 \times 2$

$\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$

Compute each entry by pairing a row with a column:

$c_{11} = (1)(5) + (2)(7) = 5 + 14 = 19$ $c_{12} = (1)(6) + (2)(8) = 6 + 16 = 22$ $c_{21} = (3)(5) + (4)(7) = 15 + 28 = 43$ $c_{22} = (3)(6) + (4)(8) = 18 + 32 = 50$

$= \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix}$

💡 Say it out loud: "row 1, column 1" gives the top-left; "row 2, column 1" gives the bottom-left. The row index follows $A$, the column index follows $B$.

Build the Product 🧮

Multiply $\begin{bmatrix} 2 & 0 \\ 1 & 3 \end{bmatrix} \begin{bmatrix} 4 & -1 \\ 5 & 2 \end{bmatrix}$. Find each requested entry.

1) $c_{11}$ (row 1 · column 1) $= \,?$ 2) $c_{12}$ (row 1 · column 2) $= \,?$ 3) $c_{21}$ (row 2 · column 1) $= \,?$

Concept Check 🎯

🔢 Matrix Operations

Part 4 of 5 — Special Matrices & Determinants

🔑 Big Payoff: A few special matrices and the determinant unlock the most useful application of all — solving systems of equations. This part sets that up.

Special Matrices

A square matrix has the same number of rows and columns ($n \times n$).

Identity matrix $I$ — has $1$s on the main diagonal (top-left to bottom-right) and $0$s elsewhere. It is the multiplicative "1" of matrices: $AI = IA = A$.

$I_2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, \qquad I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$

Zero matrix $O$ — all entries are $0$. It is the additive "0": $A + O = A$.

💡 The identity only "does nothing" when it's the right size to multiply $A$ — its dimension must match.

Concept Check 🎯

The Determinant of a $2 \times 2$ Matrix

For a square matrix, the determinant is a single number that tells you key facts about it. For a $2 \times 2$:

$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \;\Longrightarrow\; \det(A) = ad - bc$

Multiply the main diagonal, subtract the off-diagonal.

Worked Example

$\det\begin{bmatrix} 3 & 5 \\ 2 & 4 \end{bmatrix} = (3)(4) - (5)(2) = 12 - 10 = 2$

Worked Example (with negatives)

$\det\begin{bmatrix} 6 & -1 \\ 4 & 2 \end{bmatrix} = (6)(2) - (-1)(4) = 12 - (-4) = 16$

⚠️ Watch the signs! The formula is $ad - bc$, so a negative $b$ or $c$ can flip the sign of the subtracted term.

🔑 Why it matters: If $\det(A) = 0$, the matrix has no inverse and the related system has no unique solution. A nonzero determinant means the matrix is invertible.

Compute Determinants 🧮

Use $\det\begin{bmatrix} a & b \\ c & d \end{bmatrix} = ad - bc$.

1) $\det\begin{bmatrix} 4 & 3 \\ 1 & 2 \end{bmatrix} = \,?$ 2) $\det\begin{bmatrix} 5 & 0 \\ -2 & 3 \end{bmatrix} = \,?$ 3) $\det\begin{bmatrix} 2 & 6 \\ 1 & 3 \end{bmatrix} = \,?$

Reading the Determinant 🔽

🔢 Matrix Operations

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) describe matrices by size and entry, (2) add, subtract, and scale them, (3) multiply with the row·column rule, and (4) find determinants. Let's put it all together.

Quick Reference

⚠️ Three traps to remember: (1) add/subtract only with matching shapes, (2) matrix multiplication is not commutative, (3) for $\det$, it's $ad - bc$ — mind the signs.

Mixed Practice 🎯

One More Round 🧮

Let $A = \begin{bmatrix} 2 & -3 \\ 5 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} 4 & 0 \\ -1 & 6 \end{bmatrix}$.

1) Bottom-right entry of $A + B$ $= \,?$ 2) $\det(A) = \,?$ 3) Top-left entry of the product $AB$ (row 1 · column 1) $= \,?$

Exit Quiz ✅

Answer all three to finish the lesson.